Rich groups, weak second-order logic, and applications

In this chapter we initiate a study of first-order rich groups, i. e., groups where the first-order logic has the same power as the weak second-order logic. Surprisingly, there are quite a few finitely generated rich groups, they are somewhere inbetween hyperbolic and nilpotent groups (these are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev's problems in various groups.